Physlib.Relativity.Tensors.Conjugation.Basic

Given a tensor of rank $n$ with an index color structure $c = (c_0, c_1, \dots, c_{n-1})$, let $\bar{c}$ denote the conjugated color structure $(\text{bar } c_0, \text{bar } c_1, \dots, \text{bar } c_{n-1})$. This equivalence provides a bijection between the multi-index type for the conjugated structure, $\mathcal{I}(\bar{c})$, and the multi-index type for the original structure, $\mathcal{I}(c)$. It is defined by applying the slotwise correspondence between the basis labels of a color and its conjugate, utilizing the fact that $c_i$ and $\text{bar } c_i$ share the same set of index labels.

Let $S$ be a conjugate tensor species over a field $k$. For a tensor $t$ of rank $n$ with an index color sequence $c: \{0, \dots, n-1\} \to C$, the conjugation $S.\text{conjT } t$ is a tensor with the conjugated color sequence $\bar{c} = \text{bar} \circ c$. The components of the conjugated tensor are defined by taking the scalar conjugate (via the $\text{star}$ involution in $k$) of the original components: $$(S.\text{conjT } t)_b = \text{star}(t_{\phi(b)})$$ where $b \in \text{ComponentIdx}(\bar{c})$ is a multi-index for the conjugated structure, and $\phi: \text{ComponentIdx}(\bar{c}) \simeq \text{ComponentIdx}(c)$ is the canonical bijection (`componentReindex`) that maps multi-indices between conjugate colors. This operation is $\text{conj}$-semilinear.

Let $S$ be a conjugate tensor species over a field $k$. For any tensor $t$ with an index color sequence $c: \{0, \dots, n-1\} \to C$, let $S.\text{conjT } t$ denote the conjugated tensor with the conjugated color sequence $\bar{c} = \text{bar} \circ c$. For any multi-index $b$ for the conjugated color structure, the component of the conjugated tensor is given by the scalar conjugate (via the $\text{star}$ involution in $k$) of the component of the original tensor at the corresponding reindexed position: $$(S.\text{conjT } t)_b = \text{star}(t_{\phi(b)})$$ where $\phi: \text{ComponentIdx}(\bar{c}) \simeq \text{ComponentIdx}(c)$ is the canonical bijection (`componentReindex`) that identifies multi-indices between a color sequence and its conjugate.

Let $S$ be a conjugate tensor species over a ring $k$ with an involution $\text{star}$. For any scalar $r \in k$ and any tensor $t$ of rank $n$ with color sequence $c$, the conjugation of the scalar product $r \cdot t$ is equal to the conjugate of the scalar multiplied by the conjugation of the tensor: $$S.\text{conjT}(r \cdot t) = \text{star}(r) \cdot S.\text{conjT}(t)$$ This property demonstrates that the tensor conjugation operator is conjugate-semilinear.

Let $S$ be a conjugate tensor species. For any two tensors $t_1$ and $t_2$ of rank $n$ with index color sequence $c: \{0, \dots, n-1\} \to C$, the conjugation operation $\text{conjT}$ is additive: $$\text{conjT}(t_1 + t_2) = \text{conjT}(t_1) + \text{conjT}(t_2)$$ where the sum on the left is taken in the space of tensors with color sequence $c$, and the sum on the right is taken in the space of tensors with the conjugate color sequence $\bar{c} = \text{bar} \circ c$.

Let $S$ be a conjugate tensor species over a field $k$ with an involution $\text{star}$. Let $t$ be a tensor of rank $n$ with color sequence $c$, and $t'$ be a tensor of rank $m$ with color sequence $c'$. Given a reindexing $\sigma: \{0, \dots, n-1\} \to \{0, \dots, m-1\}$ such that $c' \circ \sigma = \bar{c}$ (where $\bar{c} = \text{bar} \circ c$), the conjugated tensor $S.\text{conjT } t$ is equal to the reindexed tensor $\text{permT } \sigma \, h \, t'$ if and only if for every multi-index $\phi$ of the conjugated structure $\bar{c}$, the scalar conjugate of the component of $t$ at the corresponding multi-index equals the $\phi$-th component of the reindexed tensor: $$S.\text{conjT } t = \text{permT } \sigma \, h \, t' \iff \forall \phi \in \text{ComponentIdx}(\bar{c}), \quad \text{star}(t_{\text{reindex}(\phi)}) = (\text{permT } \sigma \, h \, t')_\phi$$ Here, $t_{\text{reindex}(\phi)}$ denotes the component of $t$ at the multi-index in $\text{ComponentIdx}(c)$ that canonically corresponds to $\phi$, and $(\text{permT } \sigma \, h \, t')_\phi$ denotes the component of the reindexed tensor $\text{permT } \sigma \, h \, t'$ at the multi-index $\phi$.

For any natural number $n$ and any sequence of tensor index colors $c: \{0, \dots, n-1\} \to C$ in a conjugate tensor species $S$, the identity map $\text{id}$ on the indices $\{0, \dots, n-1\}$ satisfies the property $\text{IsReindexing}$ from $c$ to the double-conjugated sequence $\text{bar} \circ \text{bar} \circ c$.

Let $S$ be a conjugate tensor species. For any tensor $t$ of rank $n$ with a color sequence $c: \{0, \dots, n-1\} \to C$, applying the tensor conjugation operation $\text{conjT}$ twice returns the original tensor $t$ up to an identity reindexing. Specifically, $$S.\text{conjT}(S.\text{conjT } t) = \text{permT}(\text{id}, h, t)$$ where $\text{id}$ is the identity permutation on the indices and $h$ is the proof that the identity map is a valid reindexing from $c$ to the double-conjugated color sequence $\text{bar} \circ \text{bar} \circ c$.

In a conjugate tensor species $S$, let $c$ be a configuration of colours for the indices of a tensor. If a pair of indices $i$ and $j$ is contractible—that is, $i \neq j$ and the colour of index $j$ is the dual of the colour of index $i$ ($S.\tau(c_i) = c_j$)—then the pair $i$ and $j$ also satisfies the contraction condition for the conjugate colour configuration $\bar{c} = S.\text{bar} \circ c$, such that $i \neq j$ and $S.\tau(\bar{c}_i) = \bar{c}_j$. This property follows from the fact that the conjugation map $\text{bar}$ commutes with the variance map $\tau$.

In a conjugate tensor species $S$, let $t$ be a tensor of rank $n+2$ with a color sequence $c: \{0, \dots, n+1\} \to C$. Suppose the indices $i$ and $j$ satisfy the contraction condition $i \neq j$ and $S.\tau(c_i) = c_j$. Then, conjugating the contracted tensor is equivalent to contracting the conjugated tensor at the same indices $i$ and $j$: $$S.\text{conjT}(\text{contrT}_{i,j} t) = \text{contrT}_{i,j}(S.\text{conjT } t)$$ where $\text{contrT}_{i,j}$ denotes the contraction of the $i$-th and $j$-th slots, and $S.\text{conjT}$ denotes the tensor conjugation operation.

For a given colour $c$ in a conjugate tensor species $S$ over a commutative star-ring $k$, let $V_c$ be the associated vector space and $\bar{c}$ be the conjugate colour of $c$. The map `conjEquiv` is the conjugate-linear isomorphism $V_c \simeq_{sl} V_{\bar{c}}$ defined at the level of the species basis. Specifically, it maps a vector $v \in V_c$ to a vector in $V_{\bar{c}}$ by taking its coordinates relative to the basis of $V_c$, applying the star involution (conjugation) to each coordinate, and interpreting the results as coordinates relative to the basis of $V_{\bar{c}}$ (which shares the same index set).

In a conjugate tensor species $S$, for any colour $c$ and index $i$ of the basis of the vector space $V_c$, let $b_{c,i}$ denote the $i$-th basis vector. The conjugate-linear isomorphism $\text{conjEquiv} : V_c \to V_{\bar{c}}$ maps the basis vector $b_{c,i}$ to the corresponding basis vector $b_{\bar{c},i}$ in the space $V_{\bar{c}}$ associated with the conjugate colour $\bar{c}$, where the indices are identified via the equality of the basis index sets for $c$ and $\bar{c}$.

For a conjugate tensor species $S$ over a commutative star-ring $k$, a linear map $g: V_c \otimes_k V_{\bar{c}} \to k$ (representing a pairing between a color $c$ and its conjugate color $\bar{c}$) is **Hermitian** if for all vectors $x \in V_c$ and $y \in V_{\bar{c}}$, the following identity holds: $$g(x \otimes y) = \text{star}(g(\text{conjEquiv}^{-1}(y) \otimes \text{conjEquiv}(x)))$$ where $\text{conjEquiv}: V_c \xrightarrow{\simeq_{sl}} V_{\bar{c}}$ is the canonical conjugate-linear isomorphism defined for the species, $\text{conjEquiv}^{-1}$ is its inverse, and $\text{star}$ denotes the involution on the ring $k$.